Introduction

Residual analysis is a crucial aspect of regression diagnostics, aimed at assessing the goodness of fit of a regression model. Residuals are the differences between observed values and predicted values, providing insight into the model’s accuracy and potential issues. Visualizing residuals helps in identifying patterns, outliers, and potential violations of assumptions, leading to informed model improvements. This chapter delves into different methods of visualizing residuals and discusses their relative strengths.

Before discussing specific visualization methods, it’s important to understand the concept of residuals. Residuals are calculated as the difference between observed values (actual data points) and the predicted values generated by the regression model. Mathematically, the residual for the ith observation is given by: $$\text{Residual}_i=\text{Observed Value}_i-\text{Predicted Value}_i$$

Method of Visualizing Residuals

Visualizing residuals helps us identify any systematic patterns or deviations from randomness. Let’s explore some common methods of visualizing residuals:

1) Scatterplots of Residuals

Scatterplots are a simple and effective way to visualize residuals. In a scatterplot, each residual value is plotted against its corresponding predicted value. This plot helps us identify whether residuals are evenly spread around the horizontal line (zero residual) or if there’s a pattern indicating heteroskedasticity (uneven spread).

For example, if the scatterplot shows a fan-like shape, with residuals spreading wider as predicted values increase, it indicates heteroskedasticity. On the other hand, if residuals are randomly scattered around zero, the model’s assumptions are likely met.

2) Histograms and Density Plots

Histograms and density plots provide insights into the distribution of residuals. By plotting the frequency of different residual values, we can assess whether they follow a normal distribution. A bell-shaped curve suggests normality, while skewed distributions indicate potential issues.

For instance, if residuals have a skewed distribution with a heavy tail on one side, it could suggest the presence of outliers or non-linearity in the model.

3) Q-Q Plots and Probability Plots

Quantile-Quantile (Q-Q) plots compare the quantiles of residuals against the quantiles of a theoretical normal distribution. If the points on the plot align closely along a straight line, it suggests that residuals follow a normal distribution.

Probability plots are similar to Q-Q plots and are particularly useful for smaller sample sizes. Deviations from a straight line indicate departures from normality.